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The k-deck problem is concerned with finding the smallest positive integer S(k) such that there exist at least two strings of length S(k) that share the same k-deck, i.e., the multiset of subsequences of length k. We introduce the new problem of gapped k-deck reconstruction: For a given gap parameter s, we seek the smallest positive integer Gs(k) such that there exist at least two distinct strings of length Gs(k) that cannot be distinguished based on a "gapped" set of k-subsequences. The gap constraint requires the elements in the subsequences to be at least s positions apart within the original string. Our results are as follows. First, we show how to construct sequences sharing the same 2-gapped k-deck using a nontrivial modification of the recursive Morse-Thue string construction procedure. This establishes the first known constructive upper bound on G2(k). Second, we further improve this bound using the approach by Dudik and Schulman. 

The k-deck problem is concerned with finding the smallest positive integer S(k) such that there exist at least two strings of length S(k) that share the same k-deck, i.e., the multiset of subsequences of length k. We introduce the new problem of gapped k-deck reconstruction: For a given gap parameter s, we seek the smallest positive integer G s (k) such that there exist at least two distinct strings of length G s (k) that cannot be distinguished based on a "gapped" set of k-subsequences. The gap constraint requires the elements in the subsequences to be at least s positions apart within the original string. Our results are as follows. First, we show how to construct sequences sharing the same 2-gapped k-deck using a nontrivial modification of the recursive Morse-Thue string construction procedure. This establishes the first known constructive upper bound on G 2 (k). Second, we further improve this bound using the approach by Dudik and Schulman [6].